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G protein-coupled receptor-mediated calcium signalingin astrocytes

Maurizio de Pittà, Eshel Ben-Jacob, Hugues Berry

To cite this version:Maurizio de Pittà, Eshel Ben-Jacob, Hugues Berry. G protein-coupled receptor-mediated calciumsignaling in astrocytes. Maurizio De Pittà; Hugues Berry. Computational Glioscience, Springer,pp.115-150, 2019, Springer Series in Computational Neuroscience, 978-3-030-00817-8. 10.1007/978-3-030-00817-8_5. hal-01995850

G protein-coupled receptor-mediated calcium signaling in

astrocytes

Maurizio De PittaEPI BEAGLE, INRIA Rhone-Alpes, Villeurbanne, France

Eshel Ben-Jacob†

School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Israel

Hugues BerryEPI BEAGLE, INRIA Rhone-Alpes, Villeurbanne, France

February 12, 2018

Abstract

Astrocytes express a large variety of G protein-coupled receptors (GPCRs) which mediatethe transduction of extracellular signals into intracellular calcium responses. This transduc-tion is provided by a complex network of biochemical reactions which mobilizes a wealthof possible calcium-mobilizing second messenger molecules. Inositol 1,4,5-trisphosphate isprobably the best known of these molecules whose enzymes for its production and degra-dation are nonetheless calcium-dependent. We present a biophysical modeling approachbased on the assumption of Michaelis-Menten enzyme kinetics, to effectively describe GPCR-mediated astrocytic calcium signals. Our model is then used to study different mechanismsat play in stimulus encoding by shape and frequency of calcium oscillations in astrocytes.

1 Introduction

Calcium signaling is the most common measured readout of astrocyte activity in response tostimulation, be it by synaptic activity, by neuromodulators diffusing in the extracellular milieu,or by exogenous chemical, mechanical or optical stimuli. In this perspective, the individualastrocytic Ca2+ transient is thought, to some extent, as an integration of the triggering stimulus(Perea and Araque, 005a), and is thus regarded as an encoding or decoding of this stimulus,depending on the point of view (Carmignoto, 2000; De Pitta et al., 2013).

Multiple and varied are the spatiotemporal patterns of Ca2+ elevations recorded from as-trocytes in response to stimulation, each possibly carrying its own encoding (Bindocci et al.,2017). Insofar as different encoding modes could correspond to different downstream signal-ing, including gliotransmission and thereby regulation of synaptic function, understanding thebiophysical mechanisms underlying rich Ca2+ dynamics in astrocytes is crucial.

Calcium-induced Ca2+ release (CICR) from the endoplasmic reticulum (ER) is arguably thebest characterized mechanism of Ca2+ signaling in astrocytes (Zorec et al., 2012). It ensuesfrom nonlinear properties of Ca2+ channels which are found on the ER membrane and are gatedby the combined action of cytosolic Ca2+ and the second messenger molecule inositol 1,4,5-trisphosphate (IP3) (Shinohara et al., 2011, see also Chapters 2–4). This second messenger

†Deceased June 5, 2015.

1

molecule can be produced by the astrocyte either spontaneously or, notably, in response toactivation by extracellular insults activation of G protein-coupled receptors (GPCRs) found onthe cell’s plasma membrane (Parri and Crunelli, 2003; Panatier et al., 2011; Volterra et al.,2014). Hence, IP3 together with these receptors, can be regarded as integral components of theinterface whereby an astrocyte transduces extracellular insults into Ca2+ responses (Marinissenand Gutkind, 2001). Characterizing this interface is thus an essential step in our understandingof the emerging complexity of Ca2+ signals, and we devote this chapter to this purpose. In thefirst part of the chapter, we will present a concise framework to model intracellular IP3 signalingin astrocytes. This framework is general and can easily be extended to include additionalbiological details, such as for example, the regulation of GPCR binding efficiency by proteinkinase C. Some of the models presented in this chapter are also subjected to revision andcomparison with other astrocyte models in Chapters 16 and 18.

2 Modeling intracellular IP3 dynamics

2.1 Agonist-mediated IP3 production

G protein-coupled receptors form a large family of receptors which owe their name to theirextensively studied interaction with heterotrimeric G proteins (composed of an α, β and γ sub-unit) which undergo conformational changes that lead to the exchange of GDP for GTP, boundto the α-subunit, following receptor activation. Consequently, the Gα- and Gβγ-subunits stim-ulate enzymes thereby activating or inhibiting the production of a variety of second messengers(Marinissen and Gutkind, 2001).

Among all GPCRs, those that contain the Gαq subunit are linked with the cascade of chem-ical reactions that leads to IP3 synthesis. There, the Gαq subunit promotes activation of theenzyme pospholipase Cβ (PLCβ) which hydrolizes the plasma membrane lipid phosphatidyli-nositol 4,5-bisphosphate (PIP2) into diacylglycerol (DAG) and IP3 (Rebecchi and Pentyala,2000). Examples of such receptors expressed by astrocytes ex vivo and in vivo are the type Imetabotropic glutamate receptor 1 and 5 (mGluR1/5) (Wang et al., 2006; Sun et al., 2013), thepurinergic receptor P2Y1 (Jourdain et al., 2007; Di Castro et al., 2011; Sun et al., 2013), themuscarinic receptor mAchR1α (Takata et al., 2011; Chen et al., 2012; Navarrete et al., 2012)and the adrenergic α1 receptor (Bekar et al., 2008; Ding et al., 2013). While these receptorsbind different agonists, and likely display receptor-specific binding kinetics, they all share thesame downstream signaling pathway and therefore may be modeled in a similar fashion.

Several are the available models for Gαq-containing receptors, and the choice of what modelto use rather than another depends on the level of biological detail and the questions one isinterested in. Here our focus is on the rate of IP3 production upon activation of these receptors,so we wish to keep as simple as possible the description of the reactions that regulate theactivation of PLCβ by αq, β and γ subunits. This is possible, assuming that these reactionsare much faster than the downstream ones that result in IP3 production. In this case, a quasisteady-state approximation (QSSA) holds whereby, in the series of reactions that leads fromreceptor agonist binding to activation of PLCβ, the intermediate reactions involving the threereceptor’s subunits are at equilibrium on the time scale of the production of activated PLCβ.Accordingly, assuming that on average the receptor at rest (R) requires n molecules of an agonist(A) to promote activation of PLCβ (R*) at rate ON , we can write

R + nAON R∗ (1)

We further make another assumption: that the cascade of reactions that leads to GPCR-mediated IP3 synthesis has a Michaelis-Menten kinetics (see Appendix A.2), so the IP3 pro-

2

duction by PLCβ (Jβ) can be taken proportional to the fraction of bound receptors, definedas ΓA = [R∗]/[R]T, with [R]T –– [R] + [R*] being the total receptor concentration at the site ofIP3 production, i.e.,

Jβ = Oβ · ΓA (2)

In the above equation Oβ is the maximal rate of IP3 production by PLCβ and lumps informationon receptor surface density as well as on the size of the PIP2 reservoir. Importantly, these twoquantities may not be fixed, insofar as receptors are subjected to desensitization, internalizationand recycling, and the reservoir of PIP2 could also be modulated by cytosolic Ca2+ and IP3

(Rhee and Bae, 1997). The reader interested in modeling these aspects may refer to Lemonet al. (2003). In the following, we will assume Oβ constant for simplicity.

To seek an expression for Jβ, termination of PLCβ signaling has to be considered. With thisregard, as illustrated in Figure 1A, there are two possible pathways whereby IP3 production byPLCβ ends (Rebecchi and Pentyala, 2000). One is by reconstitution of the inactive G proteinheterotrimer, and coincides with unbinding of the agonist from the receptor, due to the intrinsicGTPase activity of the activated Gαq subunit. The other is by phosphorylation of the receptor,the Gαq subunit, PLCβ or some combination thereof by conventional protein kinases C (cPKC)(Ryu et al., 1990; Codazzi et al., 2001). This phosphorylation modulates either receptor affinityfor agonist binding, or coupling of the bound receptor with the G protein, or coupling of theactivated G protein with PLCβ, ultimately resulting in receptor desensitization (Fisher, 1995).

Denoting by cPKC* the active, receptor-phosphorylating kinase C, termination of PLCβ-mediated IP3 production can then be modeled by the following pair of chemical reactions:

R∗ΩN R + nA (3)

cPKC∗ + R∗OKR

ΩKRcPKC∗−R∗

ΩK cPKC∗ + R + nA (4)

From equations 3–4 we have:

dR∗

dt= ON [A]n[R]− ΩN [R∗]−OKR[cPKC∗][R∗] + ΩKR[cPKC∗−R∗] (5)

d[cPKC∗−R∗]

dt= OKR[cPKC∗][R∗]− (ΩKR + ΩK)[cPKC∗−R∗] (6)

Assuming that production of the intermediate kinase-receptor complex is at quasi steady statein reaction 4, i.e. d[cPKC∗−R∗]/dt ≈ 0, provides (equation ??)

[cPKC∗−R∗] =OKR

ΩKR + ΩK[cPKC∗][R∗] (7)

Then, substituting this latter equation in equation 5 gives

dR∗

dt= ON [A]n[R]− ΩN [R∗]−OKR

(1− ΩKR

ΩKR + ΩK

)[cPKC∗][R∗]

= ON [A]n[R]− ΩN [R∗]−OK [cPKC∗][R∗] (8)

where we defined OK = OKR (1− ΩKR/ (ΩKR + ΩK)).To retrieve an equation for [cPKC*], we consider the fact that activation of cPKC requires

binding to the kinase of free cytosolic Ca2+ (C) and DAG, but only if Ca2+ binds first, cPKC canget sensibly activated by DAG (Oancea and Meyer, 1998). Accordingly, the following sequentialbinding reaction scheme for cPKC activation may be assumed:

cPKC + Ca2+ OKC

ΩKCcPKC ′ (9)

3

cPKC ′ + DAGOKD

ΩKDcPKC∗ (10)

where cPKC is the inactive kinase, and cPKC ′ denotes the Ca2+-bound kinase complex.By QSSA in reaction 4 it follows that the available activated kinase approximately equalsto [cPKC*]T = [cPKC*] + [cPKC* –R*]≈ [cPKC*]. Moreover, it can be assumed that onlya small fraction of cPKC ′ is bound by DAG so that [cPKC∗] [cPKC ′]. In this fashion,the available cPKC, denoted by [cPKC]T, can be approximated by [PKC]T≈ [PKC] + [PKC ′].Accordingly, solving reactions 9 and 10 for [PKC∗] provides

[cPKC∗] =([cPKC∗] + [cPKC ′]

)· H1 ([DAG],KKD)

≈ [cPKC ′] · H1 ([DAG],KKD)

= [cPKC]T · H1 (C,KKC) · H1 ([DAG],KKD) (11)

where KKD = ΩKD/OKD and KKC = ΩKC/OKC , and H1 (x,K) denotes the Hill functionx/(x + K) (Appendix A.1). In practice the activation of the kinase consists of two sequentialtranslocations to the plasma membrane of its C2 and C12 domains (Oancea and Meyer, 1998).The translocation of C2 is regulated by Ca2+ whereas that of C12 is by DAG. In this processhowever, experiments showed that the initial translocation of C2 is the rate limiting step forkinase activation (Shinomura et al., 1991), inasmuch as C12 translocation rapidly follows thatof C2 (Codazzi et al., 2001). This agrees with the notion that the cPKC affinity for DAGis regarded to be much higher than the affinity of the kinase for Ca2+, i.e. KKD KKC

(Nishizuka, 1995). Since the product of two Hill functions with widely separated constantscan be approximated by the Hill function with the largest constant (De Pitta et al., 2009),equation 11 can be rewritten as

[cPKC∗] ≈ [cPKC]T · H1 (C,KKC) (12)

which, once replaced in equation 8, gives:

d[R∗]

dt= ON [A]n[R]− ΩN

(1 +

OK [cPKC]TΩN

H1 (C,KKC)

)[R∗] (13)

Finally, dividing both left and right terms in the above equation by [R]T, equation 13 can berewritten as

dΓAdt

= ON [A]n (1− ΓA)− ΩN (1 + ζ · H1 (C,KKC)) ΓA (14)

where ζ = OKC [cPKC]T/ΩN quantifies the maximal receptor desensitization by cPKC. In theapproximation that receptor binding and activation is much faster than the effective PLCβ-mediated IP3 production, ΓA can be solved for the steady state. In this fashion, IP3 productionby PLCβ in equation 2 becomes

Jβ = Oβ · Hn(

[A], (KN (1 + ζH1 (C,KKC)))1n

)(15)

where KN = ΩN/ON . The Hill coefficient n denotes cooperativity of the binding reaction ofthe agonist with the receptor and is both receptor and agonist specific. For example, glutamatebinding to subtype 1 mGluRs, such as those expressed by astrocytes (Gallo and Ghiani, 2000),is characterized by negative cooperativity and found in association with a Hill coefficient ofn = 0.48−0.88 (Suzuki et al., 2004). On the contrary, binding of ATP to P2Y1Rs of dorsal spinalcord astrocytes from rats is characterized instead by almost no cooperativity and n = 0.9 − 1(Fam et al., 2000).

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2.2 IP3 production by receptors with α subunits other than q-type

A series of other astrocytic GPCRs, that traditionally associate with non-αq subunits, havealso been reported to mediate IP3-triggered CICR, both in situ and in vivo. These includeGαi/o-coupled GABAB receptors (Kang et al., 1998; Serrano et al., 2006; Mariotti et al., 2016),endocannabinoid CB1 receptors (Navarrete and Araque, 2008; Min and Nevian, 2012), adenosin-ergic A1 receptors (Cristovao-Ferreira et al., 2013), adrenergic α2 receptors (Bekar et al., 2008),and dopaminergic D2/3 receptors (Jennings et al., 2017); as well as Gαs-coupled receptorslike adenosine A2A receptors (Cristovao-Ferreira et al., 2013), and dopamine D1/5 receptors(Jennings et al., 2017). αi/o and αs subunits are not expected to be linked with IP3 synthe-sis (Marinissen and Gutkind, 2001), rather they respectively inhibit or stimulate intracellularproduction of cAMP. Therefore the mechanism whereby these receptors could also promotemobilization of Ca2+ from IP3-sensitive ER stores remains a matter of investigation.

One obvious possibility is that some of these receptors could be atypical in astrocytes and alsobe coupled with Gαq, as it seems the case for example of astrocytic CB1Rs in the hippocampus(Navarrete and Araque, 2008) and in the basal ganglia (Martın et al., 2015). Biased agonismcould also be another possibility since the spatiotemporal pattern of agonist action on GPCRscould be quite different depending on agonist-binding kinetics of the receptor, especially ifagonists differentially engage dynamic signalling and regulatory processes (Overington et al.,2006), such as in the likely scenario of synapse-astrocyte interactions (Heller and Rusakov,2015). However, there is not yet direct structural evidence for distinct receptor conformationslinked to specific signals such as distinct G protein classes, and future studies are requiredto compare crystal structures of astrocytic GPCRs bound to biased and unbiased ligands toestablish these relationships (Violin et al., 2014).

Alternatively, other signaling pathways mediated by cAMP that result in CICR could alsobe envisaged. In particular, Doengi et al. (2009) reported that GABA-evoked astrocytic Ca2+

events in the olfactory bulb are fully prevented by blockers of astrocytic GABA transporters(GATs), but only partially by GABAB antagonists. GAT activation leads to an increase ofintracellular Na+, since this ion is cotransported with GABA, and such increase indirectlyinhibits the Na+/Ca2+ exchanger on the plasma membrane. In turn, the ensuing Ca2+ increasecould be sufficient to induce Ca2+ release from internal stores by stimulation of endogenous IP3

production (Losi et al., 2014, see the following Section). This possibility is further corroboratedby the observation that astrocytic GATs could indeed be inhibited or stimulated respectivelyby A1Rs or A2ARs (Cristovao-Ferreira et al., 2013).

Yet other mechanisms could be at play for different receptors. Dopaminergic receptors forexample could either increase (D1/5 receptors) or decrease (D2/3 receptors) intracellular Ca2+

levels in astrocytes (Jennings et al., 2017). This could indeed be explained assuming a possibleaction of these receptors on GATs which, similarly to adenosinergic receptors, could respectivelyincrease or reduce GABA/Na+ cotransport into the cell, ultimately promoting or inhibitingCICR according to what was suggested for GABABRs. However there is also evidence thatnontoxic levels of dopamine could be metabolized by monoamine-oxidase in cultured astrocytes,resulting in the production of hydrogen peroxide (Vaarmann et al., 2010). This reactive oxygenspecies ultimately activates lipid peroxidation in the neighboring membranes which in turntriggers PLC-mediated IP3 production and CICR. Overall these different scenarios unraveladditional complexity in the possible mechanisms of GPCR-mediated CICR in astrocytes andcall for future modeling efforts that are beyond the scope of this chapter.

5

2.3 Endogenous IP3 production

Phospholipase Cδ (PLCδ) is the enzyme responsible of endogenous IP3 production in astrocytes,that is IP3 production that does not require external (i.e. exogenous) stimulation (Ochocka andPawelczyk, 2003; Suh et al., 2008). The specific catalytic activity of this enzyme in the presenceof cytosolic Ca2+ is 50- to 100-fold greater than Ca2+-stimulated activity of PLCβ in theabsence of activating G protein subunits (Rebecchi and Pentyala, 2000), suggesting that PLCδ

is prominently activated by increases of intracellular Ca2+ (Rhee and Bae, 1997).Figure 1B exemplifies the biochemical network associated with PLCδ activation. Structural

and mutational studies of PLCδ complexes with Ca2+ and IP3, revealed complex interactionsof Ca2+ with several negatively charged residues within the PLCδ catalytic domain (Essen et al.,1996, 1997; Rhee and Bae, 1997), hinting cooperative binding of at least two Ca2+ ions with thisenzyme (Essen et al., 1997). In agreement with these experimental findings, we model PLCδ-mediated IP3 production (Jδ) as (Pawelczyk and Matecki, 1997; Hofer et al., 2002):

Jδ = Jδ(I) · H2 (C,Kδ) (16)

where H2 (C,Kδ) denotes the Hill function of C with coefficient 2 and affinity Kδ (Appendix B),and Jδ(I) is the maximal rate of IP3 production by PLCδ which depends on intracellular IP3

(I). Experiments revealed that high IP3 concentrations, i.e. > 1 µm, inhibit PLCδ activityby competing with PIP2 binding to the enzyme (Allen and Barres, 2009). Accordingly, themaximal PLCδdependent IP3 production rate can be modeled by

Jδ(I) =Oδ

1 + Iκδ

= Oδ (1−H1 (I, κδ)) (17)

where Oδ is the maximal rate of IP3 production by PLCδ and κδ is the inhibition constantof PLCδ activity.

2.4 IP3 degradation

There are two pathways for IP3 degradation in astrocytes. The first one is by dephosphory-lation of IP3 by inositol polyphosphate 5-phosphatase (IP-5P). The other one occurs throughphosphorylation of IP3 by the IP3 3-kinase (IP33K). Both pathways could be Ca2+ dependentbut in opposite ways: while the activity of IP33K is stimulated by cytosolic Ca2+ (Communiet al., 1997), IP-5P is inhibited instead (Communi et al., 2001) (Figure 2A). Thus, dependingon the Ca2+ concentration in the cytoplasm, different mechanisms of IP3 degradation couldexist (Sims and Allbritton, 1998). Moreover, IP-5P-mediated IP3 degradation could also beinhibited by competitive binding of inositol 1,3,4,5-tetrakisphosphate (IP4) produced by IP3-3K-mediated IP3 phosphorylation (Connolly et al., 1987; Erneux et al., 1998), thereby makingthe two degradation pathways interdependent (Hermosura et al., 2000). However, we will notconsider this aspect any further, since modeling of this reaction pathway requires a detailedconsideration of the complex metabolic network underpinning degradation of the large familyof inositol phosphates (Communi et al., 2001; Irvine and Schell, 2001). The reader interested inthese aspects may refer to Dupont and Erneux (1997) for a sample modeling approach to theproblem.

Both IP-5P-mediated dephosphorylation (J5P ) and IP33K-mediated phosphorylation of IP3

(J3K) can be described by Michaelis-Menten kinetics (Irvine et al., 1986; Togashi et al., 1997),i.e.,

J5P = J5P · H1 (I,K5) (18)

6

J3K = J3K(C) · H1 (I,K3) (19)

Since K5P > 10 µm (Verjans et al., 1992; Sims and Allbritton, 1998), and such high IP3 con-centrations are unlikely to be physiological (Lemon et al., 2003; Kang and Othmer, 2009), theactivity of IP-5P can be assumed far from saturation. Accordingly, the IP3 degradation rateby IP-5P can be linearly approximated by (Stryer, 1999):

J5P ≈ Ω5P · I (20)

where Ω5P = J5P /K5 is the maximal rate of IP-5P-mediated IP3 degradation in the linearapproximation.

IP3 phosphorylation by IP33K is regulated in a complex fashion (Figure 2A). For rest-ing conditions, when intracellular IP3 and Ca2+ concentrations are below 0.1 µM, (Parpuraand Haydon, 2000; Mishra and Bhalla, 2002; Kang and Othmer, 2009), it is very slow. On theother hand, as Ca2+ increases, IP33K activity is substantially stimulated by its phosphorylationby CaMKII in a Ca2+/calmodulin (CaM)–dependent fashion (Communi et al., 1997). A furtherpossibility could eventually be that IP33K is also inhibited by Ca2+-dependent PKC phosphory-lation (Sim et al., 1990), however, since evidence for the existence of such inhibitory pathway iscontradictory (Communi et al., 1995), this possibility will not be taken into further considerationin this study.

Phosphorylation of IP33K by active CaMKII (i.e. CaMKII*) only occurs at a single threonineresidue (Communi et al., 1997, 1999), so that it can be assumed that the rate of IP33K phos-phorylation is J∗3K(C) ∝ [CaMKII∗]. On the other hand, activation of CaMKII is Ca2+/CaM-dependent and occurs in a complex fashion because of the unique structure of this kinase, whichis composed of ∼12 subunits, with three to four phosphorylation sites each (Kolodziej et al.,2000). Briefly, Ca2+ increases lead to the formation of a Ca2+ –CaM complex (CaM+) that mayinduce phosphorylation of some of the sites of each CaMKII subunit. However, only when twoof these sites at neighboring subunits are phosphorylated, CaMKII quickly and fully activates(Hanson et al., 1994). Despite the multiple CaM+ binding reactions in the inactive kinase,experiments showed that KII activation by CaM+ can be approximated by a Hill equationwith unitary coefficient (De Konick and Schulman, 1998). Hence, the following kinetic reactionscheme for CaMKII phosphorylation can be assumed:

4 Ca2+ + CaMO0

Ω0

CaM+ (21)

KII + CaM+ Ob

ΩbCaMKII

Ωa

ΩiCaMKII∗ (22)

Consider then first the binding reaction in 22. Assuming that the second step is very rapidwith respect to the first one (Thiel et al., 1988; De Konick and Schulman, 1998), the generationof CaMKII* is in equilibrium with CaMKII consumption, i.e.,

[CaMKII∗] ≈ Ωa

Ωi[CaMKII] (23)

Then, under the hypothesis of quasi-steady state for CaMKII,

d[CaMKII]

dt= Ob [KII][CaM+]− (Ωa + Ωb) [CaMKII] + Ωi [CaMKII∗] ≈ 0 (24)

Replacing [CaMKII∗] from equation 23 in the latter equation provides

[CaMKII∗] = KaKb[KII][CaM+] (25)

7

where Ka = Ωa/Ωi and Kb = Ob/Ωb. Defining the total kinase II concentration as[KII]T = [KII] + [CaMKII] + [CaMKII*] and assuming it constant, equation 25 can be rewrittenas

[CaMKII∗] =Ka[KII]T1 +Ka

· H1

([CaM+],Km

)(26)

with Km = (Kb (1 +Ka))−1.

The substrate concentration for the enzyme-catalyzed reaction 22 is provided by reaction 21and reads (by QSSA)

[CaM+] = [CaM] · H4 (C,K0) (27)

with K0 = O0/Ω0. Therefore, replacing the latter expression for [CaM+] in equation 26, finallyprovides

[CaMKII∗] =Ka[KII]T1 +Ka

(1 +

Km

[CaM]

)−1

· H4

(C,

K0Km

Km + [CaM]

)(28)

Defining the Ca2+ affinity constant of IP33K as KD = K0Km/ (Km + [CaM]), the above cal-culations show that, despite its complexity, the reaction cascade underlying the activationof CaMKII can be concisely described by a Hill function of the Ca2+ concentration (C) sothat [CaMKII∗] ∝ H4 (C,KD). Accordingly, it is also J3K(C) ∝ H4 (C,KD), and equation 19for IP33K-mediated IP3 degradation can be rewritten as

J3K = O3K · H4 (C,KD)H1 (I,K3) (29)

where O3K is the maximal rate of IP3 degradation by IP33K.

3 Encoding of stimulation by combined IP3 and Ca2+ dynamics

3.1 The G-ChI model for IP3/Ca2+ signaling

A corollary of the biological and modeling arguments exposed in the previous section is thatCa2+ and IP3 signals are, generally speaking, dynamically coupled in astrocytes. This impliesthat a complete model that mimics astrocytic IP3 signaling must also include a description ofCICR. An example of such models is the so-called ChI model originally introduced by De Pittaet al. (2009), which is constituted by three ODEs respectively for intracellular Ca2+ (C), theIP3R gating variable h and the mass-balance equation for intracellular IP3 lumping terms, (16),(20) and (29), i.e.

dC

dt= Jr(C, h, I) + Jl(C)− Jp(C) (30)

dh

dt= Ωh(C, I) (h∞(C, I)− h) (31)

dI

dt= OδH2 (C,Kδ) (1−H1 (I, κδ))−O3K H4 (C,KD)H1 (I,K3)− Ω5P I (32)

The above model can be extended to explicitly modeling of GPCR dynamics by a G-ChI model.To this aim, we add to the right-hand side of equation 32 the contribution of GPCR-mediatedIP3 synthesis given by equation 15. However, if one is interested in how GPCR kinetics evolveswith IP3 and Ca2+ dynamics, then the formula for Jβ given by equation 2 must be used instead

8

of equation 15. Accordingly, the above system of equations must be completed by equation 14for astrocytic receptor activation, i.e.

dΓAdt

= . . . (14)

dC

dt= . . . (30)

dh

dt= . . . (31)

dI

dt= OβΓA +OδH2 (C,Kδ) (1−H1 (I, κδ))−O3K H4 (C,KD)H1 (I,K3)− Ω5P I (33)

Regarding the differential equations for the variables C and h above, the original formulationof the G-ChI model considered the Li-Rinzel description for CICR previously introduced inChapter 3 (Li and Rinzel, 1994). In the following, we will refer to this formulation. In practicehowever, it must be noted that any suitable model of Ca2+ and IP3R dynamics discussed inChapters 2, 3 and 16 can be adopted in lieu of the Li-Rinzel description, and accordinglydifferent models of G-ChI type may be developed, each possibly customized to study specificaspects of coupled IP3 and Ca2+ signaling in astrocytes.

Figure 3 illustrates some characteristics of IP3 and Ca2+ dynamics reproduced by the G-ChI model. In the left panel of this figure, IP3R kinetic parameters are chosen to fit, as closelyas possible, experimental data points for the steady-state open probabilities of type-2 IP3Rs atfixed Ca2+ (solid line) and IP3 concentrations (dashed line). In the right panel, the remainderof the parameters of the model are then set to reproduce (solid black line) a sample Ca2+ traceimaged by confocal microscopy on cultured astrocytes (gray data points). It may be observedhow the associated IP3 and h oscillations predicted by the model, are almost out of phase withrespect to the Ca2+ ones. For h, this is due to IP3R kinetics, whereby an increase of cytosolicCa2+ promotes receptor inactivation. For IP3 instead, this dynamics is a direct consequence ofthe Ca2+-dependent rate of degradation of this molecule by the IP33K enzyme. This is a crucialaspect of intracellular IP3 regulation in astrocytes which is addressed more in detail below.

3.2 Different regimes of IP3 signaling

To develop the G-ChI model in Section 2, we stressed on the molecular details of the Ca2+

dependence of the different enzymes involved in IP3 signaling, yet how this dependence shapesCa2+ and IP3 oscillations remains to be elucidated. With this purpose, we consider in Figure 4the simple scenario of Ca2+ oscillations triggered by repetitive stimulation of an astrocyte bypuffs of extracellular glutamate (top three panels), and look at the different contributions to IP3

production and degradation underpinning the ensuing Ca2+ and IP3 dynamics (lower panels).With this regard, it may be noted how the total rate of IP3 production (dashed line in thefourth panel from top) almost resembles the dynamics of activation of astrocyte receptors (ΓA,second panel from top) except for little bumps in correspondence of Ca2+ pulse-like elevations(solid trace, third panel from top). Consideration of the different contributions to IP3 by PLCβ

(orange trace) and PLCδ (blue trace) reveals that, while most of IP3 production is driven bymGluR-mediated PLCβ activation, those bumps are instead caused by PLCδ, whose activationis substantially boosted during intracellular Ca2+ elevations.

Similar arguments also hold for IP3 degradation (bottom panel). In this case, the total rateof IP3 degradation (dashed line) closely mimics IP3 dynamics in between Ca2+ elevations (greentrace, third panel from top), and is mostly contributed by Ca2+-independent IP-5P-mediateddegradation (violet trace). This scenario however changes during Ca2+ elevations, when IP33K

9

activation becomes significant and promotes faster rates of IP3 degradation, as mirrored by thedashed line which peaks in correspondence of Ca2+ oscillations.

Overall, these observations suggest that Ca2+-independent activity of PLCβ and IP-5P vs.Ca2+-dependent activation of PLCδ and IP33K account for different regimes of IP3 signaling.One regime corresponds to low intracellular Ca2+ close to resting concentrations, whereby IP3

is mainly produced by receptor-mediated activation of PLCβ against degradation by IP-5P.The other regime significantly adds to the former for sufficiently high Ca2+ elevations, whereIP3 production is boosted by PLCδ, but also IP3 degradation is faster by IP33K activation.

The contribution to IP3 production and degradation by each enzyme clearly depends ontheir intracellular expression as reflected by the values of the rate constants Oβ, Oδ, O3K andΩ5P in equation 33. Nonetheless, it should be noted that the existence of different regimes ofIP3 production and degradation is regardless of these rate values, insofar as it is set by thevalues of the Michaelis-Menten constants of the underpinning reactions, mostly Kδ and KD.Remarkably, estimates of these two constants are in the range of 0.1 − 1.0 µm, that is wellwithin the range of Ca2+ elevations expected for an astrocyte, whose average resting Ca2+

concentration is reported to be < 0.15 µm (Zheng et al., 2015). This assures that activation ofPLCδ and IP33K is effective only when intracellular Ca2+ approaches to, or increases beyondKδ and KD, as expected by the occurrence of CICR.

3.3 Signal integration

The existence of different regimes of IP3 signaling shapes the time evolution of IP3 with respectto stimulation in a peculiar fashion. From Figure 4 (third panel), it may indeed be notedthat, starting from resting values, IP3 increases for each glutamate puff almost stepwise, till itreaches a peak (or threshold) concentration (normalized to ∼ 1) that triggers CICR, therebytriggering a Ca2+ pulse-like elevation. This Ca2+ elevation promotes IP3 degradation to someconcentration between its peak and baseline values, in a sort of reset mechanism, leaving IP3

to increase back again to the CICR threshold until the next elevation. In between each Ca2+

elevation, counting from the first one ending at t ≈ 4 s, we may appreciate how IP3 increasesalmost proportionally to the number of glutamate puffs, akin to an integrator of the stimulus.

This may readily be proved by analytical arguments approximating, for simplicity, eachglutamate puff occurring at tk by a Dirac’s delta δ(t−tk), so that the external stimulus impingingon the astrocyte is modeled by Y (t) = G ·∆

∑k δ(t− tk), where G ·∆ represents the glutamate

concentration delivered in the time unit per puff (i.e. its dimensions are µm · s). Then, assumingthat in between oscillations, intracellular Ca2+ concentration is close to basal levels, i.e. C ≈ C0,with C0 < ()KKC , Kδ, K3 and h ≈ h∞, it is possible to reduce equations 14 and 33 to

dΓAdt≈ −(ONY (t) + ΩN )ΓA +ONY (t) (34)

dI

dt≈ −J5P + Jβ = −Ω5P I +OβΓA (35)

Using the fact that for puffs delivered at rate ν the identity∫ t ′′t ′∑

k δ(t − tk)dt = ν(t ′′ − t ′)holds, we can solve equation 34 for ΓA obtaining

ΓA(t) =

∫ t

−∞ONY (t ′) e−

∫ tt ′ (ΩN+ONY (t ′′))dt ′′dt ′

=

∫ t

−∞ONY (t ′) e−ΩN (t−t ′) e−ON

∫ tt ′ Y (t ′′)dt ′′dt ′

=

∫ t

−∞ONY (t ′) e−(ΩN+ONG∆ν)(t−t ′)dt ′

10

= ONY (t ′) ∗ ZΓA(t) (36)

where “∗” denotes the convolution operator. It is thus apparent that the fraction of acti-vated receptors ΓA(t) is an integral transform of the stimulus Y (t) by convolution with thekernel ZΓA(t). Specifically, ZΓA(t) may be regarded as the fraction of astrocyte receptors stim-ulated by one extracellular glutamate puff – or equivalently, by synaptic release triggered byan action potential –, and characterizes the encoding of the stimulus by the astrocyte via itsactivated receptors.

The IP3 signal resulting from the activated receptors then evolves according to

I(t) =

∫ t

−∞OβΓA(t ′) e−

∫ tt ′ Ω5P dt ′′dt ′ =

∫ t

−∞OβΓA(t ′) e−Ω5P (t−t ′)dt ′

= OβΓA(t) ∗ ZI(t) (37)

That is the IP3 signal is also an integral transform of the input stimuli through the fractionof activated receptors ΓA(t), by convolution with the kernel ZI(t) = e−Ω5P t. In particular,experimental evidence hints that the rate constant Ω5P is often small compared to the rate ofincoming stimulation (Appendix B), so that ZI(t) ≈ 1. In this case then, equation 37 predictsthat I(t) ≈

∫ t−∞OβΓA(t ′)dt ′, namely that the IP3 signal effectively corresponds to the integral

of the fraction of activated astrocyte receptors.

It is also worth understanding the nature of the threshold concentration that IP3 must reachin order to trigger CICR. In the G-ChI model, based on the Li-Rinzel description of CICR,this threshold may be not well-defined and generally varies with the parameter choice as wellas with the shape and amplitude of the delivered stimulation (De Pitta et al., 2009). Considerfor example Figure 5A where the Ca2+ response of an astrocyte (bottom panel) is simulatedfor different color-coded step increases of extracellular glutamate (top panel). It may be notedthat CICR, reflected by one or multiple Ca2+ pulse-like increases, is triggered by glutamateconcentrations greater or equal to the orange trace. However, the IP3 threshold for CICR(central panel) appears to grow with the extracellular glutamate concentration. This is reflectedby the first ’knee’ of the IP3 curves which reaches progressively higher values of IP3 concentrationas extracellular glutamate increases from orange to lime levels. At the same time, as shownby the black dashed curve in the top panel of Figure 5B, the latency for emergence of CICRsince stimulus onset (black marks at t = 0) decreases. This can be explained by equations 34and 35, noting that, while larger glutamate concentrations promote larger receptor-mediatedIP3 production, this increased production is also counteracted by faster degradation by IP-5P,since this latter linearly increases with IP3. Thus while larger IP3 production assures shorterdelays in the onset of CICR, a larger IP3 level must be reached to compensate for its fasterdegradation.

The top panel of Figure 5B further illustrates how the latency period for CICR onset dependson the activity of the different enzymes regulating IP3 production and degradation. Here thedifferent colored curves were obtained repeating the simulations of Figure 5A for a 50% increaseof the activity respectively of PLCβ (orange trace), PLCδ (blue trace), IP33K (red trace) andIP-5P (violet trace). In agreement with our previous analysis, PLCβ and IP-5P have the largestimpact on respectively reducing or increasing the latency period, given that they are the mainenzymes at play in IP3 signaling before CICR onset. The effect of an increase of IP3 productionby PLCδ is instead mainly significant for low glutamate concentrations, such that they couldpromote an activation of this enzyme that is comparable to that of PLCβ. Conversely, IP33Kdoes not have any role in the control of CICR latency since its activation effectively requiresCICR to onset first.

11

The variability of IP3 concentrations attained to trigger CICR by different glutamate con-centrations, and its correlation with the latency for CICR onset, suggest that the mere IP3

concentration is not an effective indicator of the CICR threshold, rather we should considerinstead the total IP3 amount produced in the astrocyte cytosol during the latency period thatprecedes CICR onset, that is the integral in time of IP3 concentration during such period. Thisis exemplified in the bottom panel of Figure 5B where such integral is plotted as a function ofthe different latency values computed in the top panel. It may be appreciated how this integralis essentially similar for different enzyme expressions (colored curves) yet associated with thesame latency value.

Taken together these results put emphasis on the crucial role exerted by IP3 signaling in thegenesis of agonist-mediated Ca2+ elevations. In particular they suggest that the expression ofdifferent enzymes responsible of IP3 production and degradation, which is likely heterogeneousacross an astrocyte, could locally set different requirements for integration and encoding ofexternal stimuli by the same cell.

3.4 Role of cPKCs and beyond

Different mechanisms of production and degradation of IP3 are only one example of the possiblemany signaling pathways that could shape the nature of Ca2+ signaling in astrocytes. There isalso compelling evidence in vitro that shape and duration of Ca2+ oscillations could be controlledby astrocyte receptor phosphorylation by cPKCs (Codazzi et al., 2001). To better understandthis aspect of astrocyte Ca2+ signaling, we relax the quasi steady-state approximation on cPKCphsophorylation and thus rewrite equation 8 as

dΓAdt

= ON [A]n (1− ΓA)− (ΩN +OKP ) ΓA (38)

where P denotes the cPKC* concentration at the receptors’ site. This in turn, requires to alsoconsider a description of cPKC* dynamics, whereby at least two additional equations in theG-ChI model must be included: one that takes into account P dynamics, but also a further onethat describes DAG dynamics (D), which is responsible for cPKC activation by Ca2+-dependenttranslocation of the inactive kinase to the plasma membrane (Oancea and Meyer, 1998).

By QSSA, the quantity of cPKC* is conserved during receptor phosphorylation in reaction 4.In this fashion, cPKC* production and degradation are only controlled by the pair of reactions 9and 10. On the other hand, taking into account from Section 2.1 that production of cPKC*

depends on the availability of the Ca2+-bound kinase complex cPKC ′, we may assume at firstapproximation that reaction 9 for Ca2+-binding to the kinase is at equilibrium, i.e. [cPKC ′] =[cPKC]TH1 (C,KKC). Accordingly, we can consider cPKC* dynamics to be driven simply byreaction 10, i.e.

dP

dt= JKP − JKD

= OKD[cPKC ′] ·D − ΩKDP

= OKD[cPKC]TH1 (C,KKC) ·D − ΩKDP

≡ OKDH1 (C,KKC) ·D − ΩKDP (39)

where we re-defined OKD ← OKD[cPKC]T as the maximal rate of cPKC* production (inµms−1).

To model DAG dynamics we start instead from the consideration that PLC isoenzymeshydrolyze PIP2 into one molecule of IP3 and one of DAG, so that DAG production coincideswith that of IP3 (Berridge and Irvine, 1989, and see also Figure 2B). Yet, only part of this

12

produced DAG is used to activate cPKC, while the rest is mainly degraded by diacylglycerolkinases (DAGKs) into phosphatidic acid (Carrasco and Merida, 2007) and, to a minor extent, bydiacylglycerol lipases (DAGLs) into 2-arachidonoylglycerol (2-AG), although this latter pathwayhas only been linked to some types of metabotropic receptors in astrocytes (Bruner and Murphy,1990; Giaume et al., 1991; Walter et al., 2004). Other pathways of use of DAG are also possiblein principle, inasmuch as DAG is a key molecule in the cell’s lipid metabolism and a basiccomponent of membranes. Nonetheless there is evidence that DAG levels are strictly regulatedwithin different subcellular compartments, and DAG generated by GPCR stimulation is notusually consumed for metabolic purposes (van der Bend et al., 1994; Carrasco and Merida,2007).

DAGK activation reflects the sequence of Ca2+mediated translocation, DAG binding andactivation that is also required for cPKCs, so the two reactions may be thought to be character-ized by similar kinetics, yet with an important difference. Sequence analysis of DAGKα, γ – thetwo isoforms of DAGKs most likely involved in astrocytic GPCR signaling (Dominguez et al.,2013) – reveals in fact the existence of two EF-hand motifs characteristics of Ca2+-binding andtwo C1 domains for DAG binding (Merida et al., 2008). In this fashion, a Hill exponent of 2instead of 1 as in equation 39 must be considered for the DAGK activating reaction, so thatDAGK-mediated DAG degradation can be modeled by

JD = ODH2 (C,KDC)H2 (D,KDD) (40)

Finally, to take into account other mechanisms of DAG degradation (JA), including but notlimited to DAGLs, we assume a linear degradation rate, i.e. JA = ΩDD. This is a crudeapproximation insofar as DAGL, could also be activated in a Ca2+-dependent fashion (Rosen-berger et al., 2007). Nonetheless, the complexity of the molecular reactions likely involved inthese other pathways of DAG degradation would require to consider additional equations in ourmodel which are beyond the scope of this chapter. The reader who is interested in these furtheraspects, may refer to Cui et al. (2016) for a possible modeling approach. For the purposes ofour analysis instead, we will consider the following equation for DAG dynamics:

dD

dt= Jβ + Jδ − JKP − JD − JA

= OβΓA +OδH2 (C,Kδ) (1−H1 (I, κδ)) +

−OKDH1 (C,KKC) ·D −ODH2 (C,KDC)H2 (D,KDD)− ΩDD (41)

Figure 6A shows a comparison of experimental Ca2+ and cPKC* traces with those repro-duced by the G-ChI model including equations 39 and 41. For inherent limitations of theLi-Rinzel description of the gating kinetics of IP3Rs, which fails to describe these receptors’open probability for large Ca2+ concentration (Figure 3) and predicts fast rates of receptorde-inactivation (O2/d2, Table D1), the G-ChI model cannot generate Ca2+ peaks as large asthose experimentally observed and shown here. Nonetheless we would like to emphasize howour model qualitatively matches experimental Ca2+-dependent cPKC* dynamics, accuratelyreproducing the phase shift between Ca2+ and cPKC* oscillations. This phase shift is criti-cally controlled by the constant KKC for Ca2+ binding to the kinase, along with the rates ofcPKC* production vs. degradation, i.e. OKD vs. ΩKD (equation 39), and the rate of receptorphosphorylation OK (equation 38).

Figure 6B further reveals the role of these rate constants in the control of Ca2+ oscillations.In this figure, we simulated the astrocyte response for a step increase of ∼ 1.5 µm extracellularglutamate, starting from resting conditions, both in the absence of kinase-mediated receptorphosphorylation (gray trace) and in the presence of it, for two different OK rate values (black

13

traces). It may be noted how receptor phosphorylation by cPKC can rescue Ca2+ oscillationsthat otherwise would vanish by saturating intracellular IP3 concentrations ensuing from largereceptor activation. This activation indeed is decreased by cPKC* according to equation 38,thereby regulating intracellular IP3 within the range of Ca2+ oscillations. Nonetheless, asthe rate of receptor phosphorylation increases (dash-dotted trace), the period of oscillationsappears to slow down and oscillations even fail to emerge, if the supply of cPKC* results ina phosphorylation rate of astrocyte receptors that exceeds their agonist-mediated activation(results not shown).

These considerations can be explained considering the period of Ca2+ oscillations as a func-tion of the extracellular glutamate concentration. As shown in Figure 6C, cPKC-mediatedreceptor phosphorylation shifts (black curves) the range of glutamate concentrations that trig-ger Ca2+ oscillations to higher values than those otherwise expected in the absence of it (graycurve). In particular, and in agreement with experimental findings (Codazzi et al., 2001), theexact value of the rate OK for receptor phopshorylation sets the entity of this shift, accountingeither for Ca2+ oscillations of period longer than without receptor phosphorylation, or for therequirement of larger glutamate concentrations to observe such oscillations. This is respectivelyreflected by the portions of the black curves that are within the range of extracellular gluta-mate concentrations of the gray curve), and those that instead are not. On the other hand,longer-period oscillations in the presence of receptor phosphorylation are likely to be observedas long as the rate of cPKC* activation by DAG (OKD) is below some critical value. A three-fold increase of this rate indeed requires glutamate concentrations beyond those needed in theabsence of receptor phosphorylation to trigger oscillations, regardless of the OK value at play(blue curves). In this scenario in fact, the large supply of cPKC*, resulting from the highOKD value, favors phosphorylation of receptors while hindering intracellular buildup of IP3 totrigger CICR. This in turn requires a larger recruitment of astrocyte receptors by larger agonistconcentrations to evoke Ca2+ oscillations.

4 Conclusions

The modeling arguments introduced in this chapter overall suggest a great richness in thepossible modes whereby astrocytes could translate extracellular stimuli into intracellular Ca2+

dynamics. These modes are brought forth by a complex network of biochemical reactions thatis exquisitely nonlinearly coupled with Ca2+ dynamics through different second messengers,among which IP3 and possibly DAG could play a paramount signaling role. In particular, theregulation of different regimes of IP3 production and degradation by Ca2+ in parallel with thedifferential regulation by this latter and DAG of the activities of cPKCs and DAGKs opens tothe scenario of the existence of different regimes of signal transduction that a single astrocytecould multiplex towards different intracellular targets depending on different local conditions ofneuronal activity.

An interesting implication emerging from our analysis of the regulation of the period ofCa2+ oscillations by cPKCs and DAG-related lipid signals is the possibility that these pathways,which could be crucially linked with inflammatory responses underpinning reactive astrocytosis(Brambilla et al., 1999; Griner and Kazanietz, 2007), could be found at different operationalstates, akin to what suggested for proinflammatory cytokines like TNFα (Santello and Volterra,2012). In our analysis for example, intermediate activation of cPKC activity could promoteCa2+ oscillations at physiological rates, while an increase of it could exacerbate fast, potentiallyinflammatory Ca2+ responses (Sofroniew and Vinters, 2010).

Similar arguments also hold for IP3 signaling. Calcium-dependent IP3 production by PLCδ

and PLCβ (via cPKC) could modulate the rate of integration of synaptic stimuli and thus dictate

14

the threshold synaptic activity triggering CICR. On the other hand, the existence of differentregimes of IP3 degradation could be responsible for different cutoff frequencies of synapticrelease, beyond which integration of external stimuli by the cells could cease. In particular, thiscutoff frequency could be mainly set by IP-5P during low synaptic activity, possibly associatedwith low intracellular Ca2+ levels, while be dependent on IP33K in regimes of strong astrocyteCa2+ activation, and thus ultimately depend on the history of activation of the astrocyte. Thefollowing chapter looks closely at some of these aspects, focusing in particular, on the role ofdifferent IP3 degradation regimes in the genesis and shaping of Ca2+ oscillations.

15

Appendix A Arguments of chemical kinetics

A.1 The Hill equation

In biochemistry, the binding reaction of n molecules of a ligand L to a receptor macromolecule R,i.e.,

R + nLkf

kb

RLn (42)

can be mathematically described by the differential equation

d[RLn]

dt= kf [R][L]n − kb[RLn] (43)

where kf , kb denote the forward (binding) and backward (unbinding) reaction rates respectively.At equilibrium,

0 = kf [R][L]n − kb[RLn]⇒ [RLn] =[R][L]n

Kd(44)

where Kd = kb/kf is the dissociation constant of the binding reaction 42. Then, the fractionof bound receptor macromolecules with respect to the total receptor macromolecules can beexpressed by the Hill equation (Stryer, 1999)

Bound

Total=

[RLn]

[R] + [RLn]=

[L]n

Kd

[L]n

Kd+ 1

=[L]n

[L]n +Kd=

[L]n

[L]n +Kn0.5

= Hn ([L],K0.5) (45)

where the function Hn ([L],K0.5) denotes the sigmoid (Hill) function [L]n / ([L]n + K0.5n), and

K0.5 = n√Kd is the receptor affinity for the ligand L, and corresponds to the ligand concentration

for which half of the receptor macromolecules are bound (i.e. the midpoint of the Hn ([L],K0.5)curve). The sigmoid shape of Hn ([L],K0.5) denotes saturation kinetics in the binding reac-tion 42, that is, for [L] K0.5 almost all the receptor molecules are bound to the ligand, sothat the fraction of bound receptor molecules does not essentially change for an increase of [L].

The coefficient n, also known as Hill coefficient, quantifies the cooperativity among multipleligand binding sites. A Hill coefficient n > 1 denotes positively cooperative binding, wherebyonce one ligand molecule is bound to the receptor macromolecule, the affinity of the latter forother ligand molecules increases. Conversely, a value of n < 1 denotes negatively cooperativebinding, namely when binding of one ligand molecule to the receptor decreases the affinity ofthe latter to bind further ligand molecules. Finally, a coefficient n = 1 denotes completelyindependent binding when the affinity of the receptor to ligand molecules is not affected by itsstate of occupation by the latter.

For unimolecular reactions, n = 1 coincides with the number of binding sites of the receptor.For multimolecular reactions involving η > 1 ligand molecules instead, the Hill coefficient ingeneral, only loosely estimates the number of binding sites, being n ≤ η (Weiss, 1997). Thisfollows from the hypothesis of total allostery that is implicit in the reaction 42, whereby the Hillfunction is a very simplistic way to model cooperativity. It describes in fact the limit case whereaffinity is 0 if no ligand is bound, and infinite as soon as one receptor binds. That is, only twostates are possible: free receptor and receptor with all ligand bound. More realistic descriptionsare available in literature, such as for example the Monod–Wyman–Changeux (MWC) model,but they yield much more complex equations and more parameters (Changeux and Edelstein,2005).

16

A.2 The Michaelis-Menten model of enzyme kinetics

The Michaelis-Menten model of enzyme kinetics is one of the simplest and best-known modelsto describe the kinetics of enzyme-catalyzed chemical reactions. In general enzyme-catalyzedreactions involve an initial binding reaction of an enzyme E to a substrate S to form a com-plex ES. The latter is then converted into a product P and the free enzyme by a further reactionthat is mediated by the enzyme itself and can be quite complex and involve several interme-diate reactions. However, there is typically one rate-determining enzymatic step that allowsthis reaction to be modeled as a single catalytic step with an apparent rate constant kcat. Theresulting kinetic scheme thus reads

E + Skf

kb

ESkcat P + E (46)

By law of mass action, the above kinetic scheme gives rise to 4 differential equations (Stryer,1999):

d[S]

dt= −kf [E][S] + kb[ES] (47a)

d[E]

dt= −kf [E][S] + kb[ES] + kcat[ES] (47b)

d[ES]

dt= kf [E][S]− kb[ES]− kcat[ES] (47c)

d[P]

dt= kcat[ES] (47d)

In the Michaelis-Menten model the enzyme is a catalyst, namely it only facilitates the reac-tion whereby S is transformed into P, hence its total concentration [E]T –– [E] + [ES] must bepreserved. This is indeed apparent by the sum of the second and the third equations above,since: d([E]+[ES])

dt = d[E]Tdt = 0⇒ [E]T = const.

The system of equations 47 can be solved for the products P as a function of the concentrationof the substrate [S]. A first solution assumes instantaneous chemical equilibrium between the

substrate S and the complex ES, i.e. d[S]dt = 0, whereby the initial binding reaction can be

equivalently described by a Hill equation (Keener and Sneyd, 2008), i.e.,

[ES]

[E]T=

[S]

[S] +Kd⇒ [ES] =

[E]T [S]

[S] +Kd(48)

Alternatively, the quasi-steady-state assumption (QSSA) that [ES] does not change on the timescale of product formation can be made, so that d

dt [ES] = 0 ⇒ kf [E][S] = kb[ES] + kcat[ES](Keener and Sneyd, 2008), and

kf [E][S] = kb[ES] + kcat[ES]⇒ kf ([E]T − [ES]) [S] = kb[ES] + kcat[ES]

⇒ kf [E]T [S] = (kf [ES][S] + kb[ES] + kcat[ES])

⇒ [ES] = [E]T[S]

[S] +KM(49)

where KM = (kb + kcat) /kf is the Michaelis-Menten constant of the reaction which quantifiesthe affinity of the enzyme to bind to the substrate.

Regardless of the hypothesis made to find an expression for [ES], the rate vP of productionof P can be always written as

vP =d[P]

dt= kcat[ES] = kcat[E]T

[S]

[S] +K0.5= vmax

[S]

[S] +K0.5(50)

17

where vmax = kcat[E]T is the maximal rate of production of P in the presence of enzyme satu-ration, when all the available enzyme takes part in the reaction; and the affinity constant K0.5

equals the dissociation constant Kd of the initial binding reaction in the chemical equilibriumapproximation (equation 48), or the Michaelis-Menten constant in the QSSA (equation 49).

An important corollary of the Michaelis-Menten model of enzyme kinetics is that the fractionof the total enzyme that forms the intermediate complex ES can be expressed by a Hill equationof the type

[ES]

[E]T=

[S]

[S] +K0.5= H1 ([S],K0.5) (51)

and K0.5 can be regarded as the half-saturating substrate concentration of the reaction. Simi-larly, the effective reaction rate vP (equation 51) is proportional to the maximal reaction rateby a Hill-like term H1 ([S],K0.5).

Appendix B Parameter estimation

B.1 Metabotropic receptors

Rate constants ON , ΩN (equation 14) lump information on astrocytic metabotropic receptors’activation and inactivation, namely how long it takes for these receptors, once bound by theagonist, to trigger PLCβ-mediated IP3 production and how long this latter lasts. Since IP3

production mediated by agonist binding with the receptors controls the initial intracellularCa2+ surge, these two rate constants may be estimated by rise times of agonist-triggered Ca2+

signals. With this regard, experiments reported that application of 50 µm DHPG – a potentagonist of mGluR5 which are the main type of metabotropic glutamate receptors expressedby astrocytes (Aronica et al., 2003) –, triggers submembrane Ca2+ signals characterized by arise time τr = 0.272 ± 0.095 s. Because mGluR5 affinity (K0.5) for DHPG is ∼ 2 µm (Brabetet al., 1995), that is much smaller than the applied agonist concentration, receptor saturationmay be assumed in those experiments whereby the receptor activation rate by DHPG (ODHPG)can be expressed as a function of τr (Barbour, 2001), i.e. ODHPG ≈ τr/(50 µm) = 0.055 −0.113 µm−1s−1, so that ΩDHPG = ODHPGK0.5 ≈ 0.11−0.22 s−1. Corresponding rate constants forglutamate may then be estimated assuming similar kinetics, yet with K0.5 = KN = ΩN/ON ≈3 − 10 µm (Daggett et al., 1995), that is 1.5–5-fold larger than K0.5 for DHPG. Moreover,since rise times of Ca2+ signals triggered by non-saturating physiological stimulation are fasterthan in the case of DHPG (Panatier et al., 2011), it may be assumed that ON > ODHPG.With this regard, for a choice of ON ≈ 3 × ODHPG = 0.3 µm−1s−1, with KN = 6 µm suchthat ΩN = (0.3 µm−1s−1)(6 µm) = 1.8 s−1, a peak of extracellular glutamate concentrationof 250 µm, delivered at t = 0 and exponentially decaying at rate Ωc = 40 s−1 (Clements et al.,1992), is consistent with a peak fraction of bound receptors of ∼ 0.75 within ∼ 70 ms fromstimulation (equation 14), which is in good agreement with experimental rise times.

B.2 IP3R kinetics

We consider a steady-state receptor open probability in the form of popen(C, I) = H31(I, d1) ·

·H31(C, d5)(1−H1 (C,Q2))3 with Q2 = d2(I + d1)/(I + d3) (see Chapter 3) and choose parame-

ters to fit corresponding experimental data by Ramos-Franco et al. (2000) for (i) different Ca2+

concentrations (C at a fixed IP3 level of I = 1 µm, i.e. p(C); and (ii) for different IP3 concentra-tions (I) at an intracellular Ca2+ concentration of C = 25 nm, i.e. p(I). To reduce the problemdimensionality while retaining essential dynamical features of IP3 gating kinetics we set d1 = d3

(Li and Rinzel, 1994). Accordingly, defining the vector parameter xp = (d1, d2, d5, O2), we min-imize the cost function cp(xp) = (popen(C, I) − p(C))2 + (popen(C, I) − p(I))2 by the Artificial

18

Bee Colony (ABC) algorithm (Karaboga and Basturk, 2007) considering 2000 evolutions of acolony of 100 individuals.

Ultrastructural analysis of astrocytes in situ revealed that the probability of ER localiza-tion in the cytoplasmic space at the soma is between ∼40–70% (Pivneva et al., 2008). Thissuggests that the corresponding ratio between ER and cytoplasmic volumes (ρA) is comprisedbetween ∼0.4–0.7.

To estimate the cell’s total free Ca2+ content CT we make the consideration that the restingCa2+ concentration in the cytosol is < 0.15 µm (Zheng et al., 2015) and can be neglectedwith respect to the amount of Ca2+ stored in the ER (CER) (Berridge et al., 2003). Hence,with CER ≥ 10 µm (Golovina and Blaustein, 1997) and a choice of ρA ≥ 0.4, it follows thatCT ≈ ρACER ≥ 4 µm. In conditions close to store depletion during oscillations (Camello et al.,2002), this latter value would also coincide with the peak Ca2+ reached in the cytoplasm, whichis reported between < 5 µm and ∼ 20 µm (Csordas et al., 1999; Parpura and Haydon, 2000;Kang and Othmer, 2009; Shigetomi et al., 2010).

In our simulations we set ρA = 0.5 while leaving arbitrary the choice of CT as far as theresulting Ca2+ oscillations qualitatively resemble the shape of those observed in experiments.The remaining parameters for CICR, i.e. zc = (ΩC , OP ), were chosen to approximate the num-ber and period of Ca2+ oscillations observed on average in experiments on cultured astrocytesthat were stimulated by glutamate perfusion. By “on average” we mean that we considered theaverage trace resulting from n = 5 different Ca2+ signals generated within the same period oftime and by the same stimulus in identical experimental conditions.

B.3 IP3 signaling

Once set the CICR parameters, individual Ca2+ traces used to obtained the above-mentioned“average trace” were used to search for zp = (Oβ, Oδ, O3K ,Ω5P ), assuming random initialconditions. The ensuing parameter values were also used in Figures 4–6 although Oβ, Oδ andO3K were increased, from case to case, by a factor comprised between 1.2− 2 either to expandthe oscillatory range or to promote CICR emergence (by increasing Oβ, Oδ) or termination (bylarger O3K values).

B.4 cPKC and DAG signaling

Calcium-dependent cPKC-mediated phosphorylation has been documented for astrocyticmGluRs and P2Y1Rs (Codazzi et al., 2001; Hardy et al., 2005) and results in a reductionof receptor binding affinity by a factor ζ ≈ 2 − 10 (Hardy et al., 2005), or possibly higher de-pending on the cell’s expression of cPKCs (Nakahara et al., 1997; Shinohara et al., 2011). Sinceexperiments showed that cPKC is robustly activated only when Ca2+ increases beyond half ofthe peak concentration reached during oscillations (Codazzi et al., 2001) then, considering peakCa2+ values of ∼ 1 − 3 µm (Shigetomi et al., 2010) allows estimating Ca2+ affinity of cPKCin the range of KKC ≤ 0.5 − 1.5 µm which indeed comprises the value of ∼ 700 nm predictedexperimentally (Mosior and Epand, 1994). Of the same order of magnitude also is the Ca2+

affinity reported for DAGK, i.e. KDC ≈ 0.3−0.4 µm (Sakane et al., 1991; Yamada et al., 1997).Reported values of DAG affinities for cPKC and DAGK may considerably differ. Micellar

assays of cPKCs activity, suggests values of KKD as low as 4.6–13.3 nm (Ananthanarayananet al., 2003), whereas studies on purified DAGK suggest a substrate affinity for this kinase ofKDD ≈ 60 µm (Kanoh et al., 1983). The differences in experimental setups and the possibilitythat the activity of these kinases could be widely regulated by different DAG pools make theseestimate of scarce utility for our model, where the DAG concentration is of the same orderof magnitude of IP3 one. With this regard we choose to set these affinities to 0.1 µm which

19

corresponded in our simulations to the average intracellular DAG concentration during Ca2+

oscillations.The remaining parameters, namely zk = (OKD, OK ,ΩD, OD,ΩD) were arbitrarily chosen

considering two constrains: (i) DAG concentration for damped Ca2+ oscillations must stabilizeto a constant value; and (ii) the down phase of cPKC* oscillations must follow that of Ca2+

ones as suggested by experimental observations by Codazzi et al. (2001).

Appendix C Software

The Python file figures.py used to generate the figures of this chapter can be downloadedfrom the online book repository at https://github.com/mdepitta/comp-glia-book. Thesoftware for this chapter is organized in two folders. The data folder contains data to fit theG-ChI model. WebPlotDigitizer 4.0 (https://automeris.io/WebPlotDigitizer) was usedto extract experimental data by Ramos-Franco et al. (2000, Figures 6 and 7) and Codazzi et al.(2001, Figure 5). The Jupyter notebook file data_loader.ipynb found in this folder containsthe code to load and clean experimental data used in the simulations.

The code folder contains instead all the routines (including figures.py) used for the simu-lations of this chapter. The two files astrocyte_models.h and astrocyte_models.cpp contains thecore G-ChI model implementation in C/C++11, while the class Astrocyte inastrocyte_models.py provides the Python interface to simulate the G-ChI model. The modelwas integrated by a variable-coefficient linear multistep Adams method in Nordsieck form whichproved robust to correctly solve stiff problems rising from different parameter choices (Skeel,1986). Model fitting is provided by gchi_fit.py and relies on the PyGMO 2.6 optimizationpackage (https://github.com/esa/pagmo2.git).

The library gchi_bifurcation.py provides routines to estimate the period and range ofCa2+ oscillation as in Figures 6. These routines use numerical continuation of the extendedG-ChI model by the Python module PyDSTool 0.92 (Clewley, 2012, https://github.com/

robclewley/pydstool).

20

Appendix D Model parameters used in simulations

Table D1. Model parameters used in the simulations, unless differently specified in figurecaptions.

Symbol Description Value Units

Astrocyte receptors

ΩN Rate of receptor de-activation 1.8 s−1

ON Rate of agonist-mediated receptor activation 0.3 µm−1s−1

n Agonist binding cooperativity 1 –

IP3R kinetics

d1 IP3 binding affinity 0.1 µmO2 Inactivating Ca2+ binding rate 0.325 µm−1s−1

d2 Inactivating Ca2+ binding affinity 4.5 µmd3 IP3 binding affinity (with Ca2+ inactivation) 0.1 µmd5 Activating Ca2+ binding affinity 0.05 µm

Ca2+fluxes

CT Total ER Ca2+ content 5 µmρA ER-to-cytoplasm volume ratio 0.5 –ΩC Maximal Ca2+ release rate by IP3Rs 7.759 s−1

ΩL Ca2+ leak rate 0.1 s−1

OP Maximal Ca2+ uptake rate 5.499 µms−1

KP Ca2+ affinity of SERCA pumps 0.1 µmIP3 production

Oβ Maximal rate of IP3 production by PLCβ 0.8 µms−1

Oδ Maximal rate of IP3 production by PLCδ 0.025 µms−1

Kδ Ca2+ affinity of PLCδ 0.5 µmκδ Inhibiting IP3 affinity of PLCδ 1.0 µm

IP3 degradation

Ω5P Rate of IP3 degradation by IP-5P 0.86 s−1

O3K Maximal rate of IP3 degradation by IP33K 0.86 µms−1

KD Ca2+ affinity of IP33K 0.5 µmK3K IP3 affinity of IP33K 1.0 µm

DAG dynamics

ΩD Unspecific rate of degradation 0.26 s−1

OD Rate of degradation by DAGK 0.45 µms−1

KDC DAGK affinity for Ca2+ 0.3 µmKDD DAGK affinity for DAG 0.1 µm

cPKC signaling

OKD Rate of cPKC* production 0.28 µms−1

ΩKD Rate of cPKC* deactivation 0.33 s−1

KKC Ca2+ affinity of PKC 0.5 µmOK Rate of receptor phosphorylation 1.0 µm−1s−1

21

A B

Figure 1. IP3 production. A Hydrolysis of the membrane lipid phosphatidylinositol 4,5-bisphosphate (PIP2) by PLCβ and PLCδ isoenzymes produces IP3 and diacylglycerol (DAG).The contribution of PLCβ to IP3 production depends on agonist binding to astrocyte G protein-coupled receptors (GPCRs). This production pathway is inhibited via receptor phosphorylationby Ca2+-dependent activation of conventional protein kinases C (cPKCs). Blue: promotingpathway; red : inhibitory pathway.

22

A

B

Figure 2. IP3 and DAG degradation. A Degradation of IP3 occurs by phosphorylation intoinositol 1,3,4,5-tetrakisphosphate (IP4) by IP33K and dephosphorylation into lower inositolphosphates by IP-5P. Both pathways are regulated by Ca2+: IP33K activity is stimulated byphosphorylation by Ca2+/calmodulin-dependent protein kinase II (CaMKII), whereas IP-5P isinhibited thereby. Moreover IP33K-mediated degradation could also be promoted by Ca2+ andDAG-dependent cPKC-mediated phosphorylation, while IP-5P could also be inhibited by IP4.For the sake of simplicity, IP-5P dependence on Ca2+ and IP4 along with IP33K dependenceon cPKC are not taken into consideration in this study (dashed pathways). B DAG is mainlydegraded into phosphatidic acid (PA) by DAG kinases (DAGK) in a Ca2+-dependent fashion,and to a minor extent, into 2-arachidonoylglycerol (2-AG) by DAG lipases (DAGL). In turn 2-AG is hydrolized by monoacylglycerol lipase (MAGL) into arachidonic acid (AA). 2-AG and AAmay promote activity of DAGK and cPKC* (orange patwhays) although this scenario is nottaken into consideration here. Colors of other pathways as in Figure 1.

23

0.01 0.1 1.0 10.0 100.0Ca2+ , IP3 (M)

0.0

0.1

0.2

0.3

0.4

0.5

IP3R

ope

n pr

obab

ility

Ca2+=25 nMIP3=1 M

0 5 10 15 20 25 30Time (s)

0.0

0.2

0.4

0.6

0.8

1.0

ChI v

ariables

(n.u.)

ChI

Figure 3. G-ChI model. (left panel) Fit of IP3Rs kinetic parameters on experimentaldata of steady-state open probabilities of type-2 IP3Rs by Ramos-Franco et al. (2000). Inthis example, and through all this chapter, we consider the Li-Rinzel description for CICR.This choice allows a reasonable fit (solid and dashed lines) of the receptors’ open probabilityas function of either intracellular IP3 (N) or intracellular Ca2+ ( ). The only exception isfor Ca2+ concentrations > 1 µm for which the open probability predicted by the Li-Rinzelmodel (solid line) vanishes much more quickly than experimental values. (right panel) SampleCa2+ (C), IP3 (I) and h traces ensuing from a simulation of the G-ChI model to reproduceexperimental Ca2+ oscillations in cultured astrocytes (gray data points) triggered by applicationof > 5 µm glutamate. Experimental data courtesy of Nitzan Herzog (University of Nottingham).A saturating glutamate concentration (i.e. ΓA = 1) was assumed with initial conditions C(0) =0.098 µm, h(0) = 0.972 and I(0) = 0.190 µm. Simulated Ca2+ and IP3 traces are reportedin normalized units with respect to minimum values of C0 = 0.1 µm and I0 = 0.16 µm andpeak values of C = 1.42 µm and I = 0.19 µm. Model parameters as in Table D1 except forOβ = 0.141 µms−1 and O3K = 0.163 µms−1.

24

Glu

(M)

0.0

0.1

0.2

0.3

Ast. Re

c.Γ A

0.0

0.5

1.0

Ca2+

, IP 3

( .u.)

CI

0.0

0.1

0.2

IP3 p

rod.

(M/s)

JβJδ

0 2 4 6 8 10Time (s)

0.0

0.1

0.2

IP3 d

egr.

(M/s)

J5PJ3K

Figure 4. Coexistence of different regimes of IP3 signaling. From top to bottom: (firstpanel) Repetitive stimulation of an astrocyte by puffs of glutamate (8 µm, rectangular pulsesat rate 0.33 Hz and 15% duty cycle); (second panel) fraction of activated astrocytic receptors;(third panel) ensuing Ca2+ (C) and IP3 (I) traces (normalized with respect to their maximumexcursion: C0 = 40 nm, I0 = 50 nm, C = 0.73 µm, I = 0.15 µm); (fourth panel) total rate ofIP3 production (dashed line) and contributions to it by PLCβ (Jβ) and PLCδ (Jδ); (bottompanel) total rate of IP3 degradation (dashed line) resulting from the combination of degradationby IP-5P (J5P ) and IP33K (J3K). Besides Ca2+ pulsed-oscillations, IP3 is mainly regulated byPLCβ (orange trace) and IP-5P (violet trace), and its concentration tends to increase in anintegrative fashion with the number of glutamate puffs. During Ca2+ elevations instead, activityof PLCδ (blue trace) and IP33K (red trace) become significant, with this latter responsible for asharp drop of intracellular IP3. Model parameters as in Table D1 except for CT = 10 µm, OP =10 µms−1 and Oδ = 0.05 µms−1.

25

A

0123

Glu

(M)

0.0

0.1

0.2

IP3 (M

)

0 1 2 3 4 5Time (s)

0

1

2

Ca2+

(M)

B

0 1 2 3 4 5Glu (M)

0

1

2

3

4

Latency (s)

referenceOβ (×1.5)Oδ (×1.5)O3K (×1.5)Ω5P (×1.5)

0 1 2 3 4Latency (s)

0.1

1

10

IP3 Time Integral (m

M⋅s)

Figure 5. Threshold for CICR. A (top panel) Step increases of extracellular glutamate (colorcoded) and resulting IP3 (central panel) and Ca2+dynamics (bottom panel) in a G-ChI astrocytemodel. Black marks at t = 0 denote stimulus onset. B (top panel) Latency for the onset ofCICR as a function of the applied glutamate concentration for the Ca2+ traces in A (blackdashed curve), as well as for 50% increases in the rate of PLCβ (Oβ), PLCδ (Oδ), IP33K (O3K)and IP-5P (Ω5P ) respectively. Emergence of CICR was detected for dC

dt ≥ 0.5 µm/s. (bottompanel) Integral of IP3 concentration as a function of the latency values computed in the toppanel. This integral is a better estimator of CICR threshold than the sole IP3 concentration.Model parameters as Figure 4.

26

A

0 35 70 105 1400.0

0.5

1.0

Expe

rimen

tal

Traces

(n.u.)

Ca2+cPKC *

0 5 10 15 20Time (s)

0.0

0.5

1.0

Simulated

Trac

es (n

.u.)

B

0.0

0.2

0.4

0.6

Ca2+

(M)

0.00.10.20.3

DAG

(M)

0 10 20 30Time (s)

0204060

cPKC

*

(nM) OK=0

OK=1.0 M−1s−1OK=3.0 M−1s−1

C

1.4 1.5 1.6 1.7 1.8 1.9Glu (M)

4

7

10

13

Oscillatio

n Pe

riod (s)

OKD=0OKD=0.28 Ms−1OKD=0.84 Ms−1

Figure 6. Regulation of Ca2+ oscillations by cPKC. A (top panel) Comparison betweenexperimental traces for Ca2+ (black) and cPKC* (red) originally recorded in cultured astrocytesby Codazzi et al. (2001) and simulations (bottom panel). Despite quantitive differences inthe shape and period of oscillations, the model can reproduce the essential correlation andphase shift between Ca2+ and cPKC* dynamics observed in experiments. Ca2+ and cPKC*

oscillations were triggered assuming an extracellular glutamate concentration of 1.48 µm, andwere normalized according to their maximum excursion: C0 = 0.04 µm, P0 = 48 nm, C = 0.49 µmand P = 65 nm. B DAG and cPKC* dynamics associated with two different rates of receptorphosphorylation by cPKC (OK , black traces) in response to a step increase of extracellularglutamate (1.55 µm at t = 0). In the absence of receptor phosphorylation (gray traces), Ca2+

oscillations would vanish due to saturating intracellular IP3 levels ensued from large receptoractivation. C Period of Ca2+ oscillations as a function of extracellular glutamate concentration.Receptor phosphorylation by cPKC critically controls the oscillatory range (black and bluecurves) with respect to the scenario without cPKC activation (gray curve). Higher glutamateconcentrations are required to trigger oscillations for larger rates of DAG-dependent cPKCactivation (OKD). Parameters as in Table D1 except for ΩC = 6.207 s−1, ΩL = 0.01 s−1, Oβ =1 µms−1.

27

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